Results & Lemmas (4)
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THEOREM 1. · coeff
THEOREM 1. For functions f e S with Hayman index a (0 < a < 1) and direction of maximal growth e iθ = 1, the inequality (4) Re £ N - 4…
THEOREM 1. For functions f e S with Hayman index a (0 < a < 1) and direction of maximal growth e iθ = 1, the inequality (4) Re £ N - 4 «4|λ.(τ.-i) holds for each N and for all λB E C, where cnm are the Grunsky coefficients and yn are the logarithmic coefficients off. This inequality is sharp for each choice of N and λn. The proof is given in §3. It is not clear whether the inequality (4) is sharp for arbitrary values of α. The method of proof suggests that an extremal value of a will be determin
Theorem 1
Theorem 1 may also be applied to derive a generalized form of the Goluzin inequalities ([5]; see [3], §4.4). We have the following result.
Theorem 1 may also be applied to derive a generalized form of the Goluzin inequalities ([5]; see [3], §4.4). We have the following result.
THEOREM 2.
THEOREM 2. For functions f e S with Hayman index a (0 < a < 1) and direction of maximal growth e ιθ = 1, the sharp inequality (5) £ f N ^Σ…
THEOREM 2. For functions f e S with Hayman index a (0 < a < 1) and direction of maximal growth e ιθ = 1, the sharp inequality (5) £ f N ^Σ -1 holds for N, for all λn e C, and for all systems of points z, e C. denotes the Koebe function.
Theorem 1
Theorem 1, we may combine Theorem 2 with the Goluzin inequalities to obtain v a
Theorem 1, we may combine Theorem 2 with the Goluzin inequalities to obtain v a